Suppose that we don't have a formula for g(x) but we know that g(1) = −3 and g'(x) = √ x^2 + 8 for all x. Use a linear approximation to estimate g(0.9) and g(1.1)?

Linearization of f(x) at a

##L(x)=f(a)+f'(a)(x-a)##

Let us find the linearization ##L(x)## of ##g(x)## at ##1##.

Since ##g(1)=-3## and ##g'(1)=sqrt{(1)^2+8}=3##,

##L(x)=g(1)+g'(1)(x-1)=-3+3(x-1)=3x-6##.

So, we can approximate:

##g(0.9) approx L(0.9)=3(0.9)-6=-3.3##

##g(1.1) approx L(1.1)=3(1.1)-6=-2.7##

I hope that this was helpful.